Rearrange the function, we have: \begin{equation}uu_{xy} = u_{x}u_{y}\end{equation}Divide both sides by \begin{equation}uu_{x}\end{equation}, we have \begin{equation}\frac{u_{xy}}{u_{x}} = \frac{u_{y}}{u}\end{equation}

which is \begin{equation} (lnu_{x})_{y} = (lnu)_{y}\end{equation}.Integrate and do some arrangements we have \begin{equation} lnu = \int{e^{f(x)}dx}+g(y) \end{equation}Thus \begin{equation} u = e^{\int{e^{f(x)}dx}}e^{g(y)} \end{equation}

Part B arrange we have \begin{equation}(lnu_{x})_{y} = u_{y} \end{equation}Integrate and arrange we have \begin{equation} -e^{-u} = \int{e^{f(x)}dx}+g(y) \end{equation}thus we have \begin{equation} u = -ln[-(\int{e^{f(x)}dx}+g(y))] \end{equation}